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User blog:Rgetar/Less-than greater-than separators in decimal and ω-ry number systems
''Note: this blog was modified on October 28, 2017. In older versions there were errors.'' First I made up <> separators for my [[Xa function]], then I realized that it may be used for any functions of multiple variables. means n separators (commas) between variables with zeros between them (zeros may be omitted). So, is n "empty" variables (zeros), if is not before a variable: <1> = ,0 <2> = ,0,0 <3> = ,0,0,0 or is n-1 "empty" variables (zeros), if is before a variable: <1> = , <2> = ,0, <3> = ,0,0, For example, let f - function of 15 variables. f(1,2,3,4,5,0,0,0,0,0,0,0,0,0,0) may be written as f(1,2,3,4,5<10>) here <10> = ,0,0,0,0,0,0,0,0,0,0 or 10 commas with 10 zeros after them f(1,3,0,0,0,0,0,0,0,0,0,0,0,0,1) may be written as f(1,3<13>1) here <13> = ,0,0,0,0,0,0,0,0,0,0,0,0, or 13 commas with 12 zeros between them Particularly, I used <> for Veblen function: Γ0 = φ(1,0,0) = φ(1<2>) φ(1,0,0,0) = φ(1<3>) φ(1<ω>) - small Veblen ordinal Multi-dimensional arrays of variables Also <> may be used to describe multi-dimensional arrays of variables: is n separators between variables (n "empty" variables if is not before a variable, or n-1 "empty" variables if is before a variable) is n separators between rows (n "empty" rows if is not before a row, or n-1 "empty" rows if is before a rows) is n separators between planes (n "empty" planes if is not before a planes, or n-1 "empty" planes if is before a planes) For example, <1,2,3> at the end of an array is three "empty" variables, two "empty" rows and one "empty" plane I used it for multi-dimensional extension Veblen function: φ(1<1,0>) - large Veblen ordinal This extension of Veblen function may be used up to Bachmann-Howard ordinal. <> in positional number systems Then I realized <> may be used in positional number systems, but with "separators" between digits instead of commas: <1> = 0 <2> = 00 <3> = 000 etc. if is not between digits or <1> = <2> = 0 <3> = 00 etc. if is between digits <1> is "separator" between digits, so, we may write <1>'s between all digits: 12345 = 1<1>2<1>3<1>4<1>5 Decimal number system In decimal number system: 1<2> = 100 1<2>1 = 101 1<6> = 1000000 1<6>1 = 1000001 1.<5>1 = 0.000001 -1<6>.<5>1 = -1000000.000001 12345<26>3<7>.<9>1<6>25 = 12345000000000000000000000000030000000.00000000010000025 1<100> - googol 1<1<100>> - googolplex 1<1<100>>1 - googolplex one ω-ry number system For infinitary number system we need infinite number of digits. For example, 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ... In ω-ry number system all natural numbers are one-digit numbers. The least two-digit number is 10 = ω More examples: 11 = ω+1 12 = ω+2 13 = ω+3 100 = ω2 101 = ω2+1 110 = ω2+ω 111 = ω2+ω+1 1000 = ω3 12300000101 = ω10+ω9·2+ω8·3+ω2+1 1<10> = ωω 1<11> = ωω+1 1<100> = ωω2 1<1<10>> = ωωω 1<1<1<10>>> = ωωωω 1<1<1<1<10>>>> = ωωωωω This may be used up to ε0. Examples with "integer ordinals": 11 = ω+1 10-1 = ω-1 -11 = -1-ω 1-10 = 1-ω Category:Blog posts